LMFDB - Newform orbit 1300.1.cy.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,1,Mod(123,1300)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(60))

chi = DirichletCharacter(H, H._module([30, 33, 25]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1300.123");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1300.cy (of order $60$, degree $16$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.648784516423$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{60})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 $ x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{60}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{60} + \cdots)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q - \zeta_{60}^{23} q^{2} - \zeta_{60}^{16} q^{4} + \zeta_{60}^{4} q^{5} - \zeta_{60}^{9} q^{8} - \zeta_{60}^{7} q^{9} +O(q^{10}) $ q - z^23 * q^2 - z^16 * q^4 + z^4 * q^5 - z^9 * q^8 - z^7 * q^9	\( q - \zeta_{60}^{23} q^{2} - \zeta_{60}^{16} q^{4} + \zeta_{60}^{4} q^{5} - \zeta_{60}^{9} q^{8} - \zeta_{60}^{7} q^{9} - \zeta_{60}^{27} q^{10} - \zeta_{60}^{8} q^{13} - \zeta_{60}^{2} q^{16} + ( - \zeta_{60}^{17} + \zeta_{60}^{6}) q^{17} - q^{18} - \zeta_{60}^{20} q^{20} + \zeta_{60}^{8} q^{25} - \zeta_{60} q^{26} + (\zeta_{60}^{19} + \zeta_{60}^{3}) q^{29} + \zeta_{60}^{25} q^{32} + ( - \zeta_{60}^{29} - \zeta_{60}^{10}) q^{34} + \zeta_{60}^{23} q^{36} + ( - \zeta_{60}^{25} + \zeta_{60}^{9}) q^{37} - \zeta_{60}^{13} q^{40} + ( - \zeta_{60}^{28} + \zeta_{60}^{21}) q^{41} - \zeta_{60}^{11} q^{45} + \zeta_{60}^{20} q^{49} + \zeta_{60} q^{50} + \zeta_{60}^{24} q^{52} + (\zeta_{60}^{22} - \zeta_{60}^{5}) q^{53} + ( - \zeta_{60}^{26} + \zeta_{60}^{12}) q^{58} + (\zeta_{60}^{29} + \zeta_{60}^{27}) q^{61} + \zeta_{60}^{18} q^{64} - \zeta_{60}^{12} q^{65} + ( - \zeta_{60}^{22} - \zeta_{60}^{3}) q^{68} + \zeta_{60}^{16} q^{72} + ( - \zeta_{60}^{19} + \zeta_{60}^{17}) q^{73} + ( - \zeta_{60}^{18} + \zeta_{60}^{2}) q^{74} - \zeta_{60}^{6} q^{80} + \zeta_{60}^{14} q^{81} + ( - \zeta_{60}^{21} + \zeta_{60}^{14}) q^{82} + ( - \zeta_{60}^{21} + \zeta_{60}^{10}) q^{85} + (\zeta_{60}^{26} + \zeta_{60}^{5}) q^{89} - \zeta_{60}^{4} q^{90} + (\zeta_{60}^{28} - \zeta_{60}^{4}) q^{97} + \zeta_{60}^{13} q^{98} +O(q^{100}) \) q - z^23 * q^2 - z^16 * q^4 + z^4 * q^5 - z^9 * q^8 - z^7 * q^9 - z^27 * q^10 - z^8 * q^13 - z^2 * q^16 + (-z^17 + z^6) * q^17 - q^18 - z^20 * q^20 + z^8 * q^25 - z * q^26 + (z^19 + z^3) * q^29 + z^25 * q^32 + (-z^29 - z^10) * q^34 + z^23 * q^36 + (-z^25 + z^9) * q^37 - z^13 * q^40 + (-z^28 + z^21) * q^41 - z^11 * q^45 + z^20 * q^49 + z * q^50 + z^24 * q^52 + (z^22 - z^5) * q^53 + (-z^26 + z^12) * q^58 + (z^29 + z^27) * q^61 + z^18 * q^64 - z^12 * q^65 + (-z^22 - z^3) * q^68 + z^16 * q^72 + (-z^19 + z^17) * q^73 + (-z^18 + z^2) * q^74 - z^6 * q^80 + z^14 * q^81 + (-z^21 + z^14) * q^82 + (-z^21 + z^10) * q^85 + (z^26 + z^5) * q^89 - z^4 * q^90 + (z^28 - z^4) * q^97 + z^13 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{4} + 2 q^{5}+O(q^{10}) $ 16 * q - 2 * q^4 + 2 * q^5	$ 16 q - 2 q^{4} + 2 q^{5} - 2 q^{13} + 2 q^{16} + 4 q^{17} - 16 q^{18} + 8 q^{20} + 2 q^{25} - 8 q^{34} - 2 q^{41} - 8 q^{49} - 4 q^{52} - 2 q^{53} - 2 q^{58} + 4 q^{64} + 4 q^{65} + 2 q^{68} + 2 q^{72} - 6 q^{74} - 4 q^{80} - 2 q^{81} - 2 q^{82} + 8 q^{85} - 2 q^{89} - 2 q^{90}+O(q^{100}) $ 16 * q - 2 * q^4 + 2 * q^5 - 2 * q^13 + 2 * q^16 + 4 * q^17 - 16 * q^18 + 8 * q^20 + 2 * q^25 - 8 * q^34 - 2 * q^41 - 8 * q^49 - 4 * q^52 - 2 * q^53 - 2 * q^58 + 4 * q^64 + 4 * q^65 + 2 * q^68 + 2 * q^72 - 6 * q^74 - 4 * q^80 - 2 * q^81 - 2 * q^82 + 8 * q^85 - 2 * q^89 - 2 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$-\zeta_{60}^{25}$	$-1$	$-\zeta_{60}^{21}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

123.1

−0.207912	−	0.978148i
−0.994522	+	0.104528i
−0.743145	+	0.669131i
0.406737	−	0.913545i
−0.406737	+	0.913545i
0.743145	−	0.669131i
0.994522	−	0.104528i
0.207912	+	0.978148i
−0.207912	+	0.978148i
−0.743145	−	0.669131i
−0.994522	−	0.104528i
−0.406737	−	0.913545i
0.406737	+	0.913545i
0.994522	+	0.104528i
0.743145	+	0.669131i
0.207912	−	0.978148i

0.994522

−

0.104528i

0.978148

−

0.207912i

0.669131

−

0.743145i

0.951057

−

0.309017i

−0.994522

−

0.104528i

0.587785

−

0.809017i

167.1

−0.743145

−

0.669131i

0.104528

0.994522i

0.913545

−

0.406737i

0.587785

−

0.809017i

0.743145

−

0.669131i

−0.951057

−

0.309017i

223.1

−0.406737

0.913545i

−0.669131

−

0.743145i

−0.978148

−

0.207912i

0.951057

−

0.309017i

0.406737

0.913545i

0.587785

−

0.809017i

267.1

−0.207912

0.978148i

−0.913545

−

0.406737i

−0.104528

0.994522i

0.587785

−

0.809017i

0.207912

0.978148i

−0.951057

−

0.309017i

383.1

0.207912

−

0.978148i

−0.913545

−

0.406737i

−0.104528

0.994522i

−0.587785

0.809017i

−0.207912

−

0.978148i

0.951057

0.309017i

427.1

0.406737

−

0.913545i

−0.669131

−

0.743145i

−0.978148

−

0.207912i

−0.951057

0.309017i

−0.406737

−

0.913545i

−0.587785

0.809017i

483.1

0.743145

0.669131i

0.104528

0.994522i

0.913545

−

0.406737i

−0.587785

0.809017i

−0.743145

0.669131i

0.951057

0.309017i

527.1

−0.994522

0.104528i

0.978148

−

0.207912i

0.669131

−

0.743145i

−0.951057

0.309017i

0.994522

0.104528i

−0.587785

0.809017i

687.1

0.994522

0.104528i

0.978148

0.207912i

0.669131

0.743145i

0.951057

0.309017i

−0.994522

0.104528i

0.587785

0.809017i

787.1

−0.406737

−

0.913545i

−0.669131

0.743145i

−0.978148

0.207912i

0.951057

0.309017i

0.406737

−

0.913545i

0.587785

0.809017i

903.1

−0.743145

0.669131i

0.104528

−

0.994522i

0.913545

0.406737i

0.587785

0.809017i

0.743145

0.669131i

−0.951057

0.309017i

947.1

0.207912

0.978148i

−0.913545

0.406737i

−0.104528

−

0.994522i

−0.587785

−

0.809017i

−0.207912

0.978148i

0.951057

−

0.309017i

1003.1

−0.207912

−

0.978148i

−0.913545

0.406737i

−0.104528

−

0.994522i

0.587785

0.809017i

0.207912

−

0.978148i

−0.951057

0.309017i

1047.1

0.743145

−

0.669131i

0.104528

−

0.994522i

0.913545

0.406737i

−0.587785

−

0.809017i

−0.743145

−

0.669131i

0.951057

−

0.309017i

1163.1

0.406737

0.913545i

−0.669131

0.743145i

−0.978148

0.207912i

−0.951057

−

0.309017i

−0.406737

0.913545i

−0.587785

−

0.809017i

1263.1

−0.994522

−

0.104528i

0.978148

0.207912i

0.669131

0.743145i

−0.951057

−

0.309017i

0.994522

−

0.104528i

−0.587785

−

0.809017i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
325.bi	even	60	1	inner
1300.cy	odd	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1300.1.cy.a	yes	16
4.b	odd	2	1	CM	1300.1.cy.a	yes	16
13.f	odd	12	1		1300.1.ct.a	✓	16
25.f	odd	20	1		1300.1.ct.a	✓	16
52.l	even	12	1		1300.1.ct.a	✓	16
100.l	even	20	1		1300.1.ct.a	✓	16
325.bi	even	60	1	inner	1300.1.cy.a	yes	16
1300.cy	odd	60	1	inner	1300.1.cy.a	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1300.1.ct.a	✓	16	13.f	odd	12	1
1300.1.ct.a	✓	16	25.f	odd	20	1
1300.1.ct.a	✓	16	52.l	even	12	1
1300.1.ct.a	✓	16	100.l	even	20	1
1300.1.cy.a	yes	16	1.a	even	1	1	trivial
1300.1.cy.a	yes	16	4.b	odd	2	1	CM
1300.1.cy.a	yes	16	325.bi	even	60	1	inner
1300.1.cy.a	yes	16	1300.cy	odd	60	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1300, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + T^{14} + \cdots + 1 $ T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$3$	$ T^{16} $ T^16
$5$	$ (T^{8} - T^{7} + T^{5} + \cdots + 1)^{2} $ (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^2
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ (T^{8} + T^{7} - T^{5} + \cdots + 1)^{2} $ (T^8 + T^7 - T^5 - T^4 - T^3 + T + 1)^2
$17$	$ T^{16} - 4 T^{15} + \cdots + 1 $ T^16 - 4T^15 + 11T^14 - 16T^13 + 16T^12 + 2T^11 + 8T^10 - 58T^9 + 177T^8 - 206T^7 + 52T^6 + 176T^5 - 204T^4 + 28T^3 + 89T^2 - 12*T + 1
$19$	$ T^{16} $ T^16
$23$	$ T^{16} $ T^16
$29$	$ T^{16} + T^{14} + \cdots + 1 $ T^16 + T^14 - 20T^12 - 46T^10 + 114T^8 + 409T^6 + 520T^4 - 4T^2 + 1
$31$	$ T^{16} $ T^16
$37$	$ T^{16} - 8 T^{14} + \cdots + 1 $ T^16 - 8T^14 + 30T^12 - 38T^10 - 21T^8 + 58T^6 + 35T^4 + 3*T^2 + 1
$41$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 - T^14 + 2T^13 + 16T^12 + 14T^11 + 32T^10 + 56T^9 + 42T^8 + 148T^7 + 283T^6 + 188T^5 + 36T^4 + 76T^3 + 86T^2 - 6T + 1
$43$	$ T^{16} $ T^16
$47$	$ T^{16} $ T^16
$53$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 - 3T^14 - 12T^13 - 8T^12 + 10T^11 + 33T^10 + 24T^9 + 15T^8 + 36T^7 - 87T^6 - 20T^5 + 142T^4 - 48T^3 + 87T^2 + 8*T + 1
$59$	$ T^{16} $ T^16
$61$	$ T^{16} - 3 T^{14} + \cdots + 1 $ T^16 - 3T^14 - 10T^12 + 32T^10 + 69T^8 + 8T^6 + 30T^4 - 7*T^2 + 1
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ T^{16} - 2 T^{14} + \cdots + 1 $ T^16 - 2T^14 + 13T^12 - 49T^10 + 150T^8 + T^6 + 523T^4 - 37T^2 + 1
$79$	$ T^{16} $ T^16
$83$	$ T^{16} $ T^16
$89$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 - 3T^14 - 6T^13 + 13T^12 + 16T^11 - 36T^10 + 12T^9 + 153T^8 + 174T^7 + 186T^6 + 118T^5 + 28T^4 - 12T^3 + 18T^2 - 4*T + 1
$97$	$ (T^{8} + 10 T^{5} + \cdots + 25)^{2} $ (T^8 + 10T^5 - 5T^4 + 25T^2 - 25T + 25)^2
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1300.cy (of order \(60\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{60}^{23} q^{2} - \zeta_{60}^{16} q^{4} + \zeta_{60}^{4} q^{5} - \zeta_{60}^{9} q^{8} - \zeta_{60}^{7} q^{9} +O(q^{10}) \) q - z^23 * q^2 - z^16 * q^4 + z^4 * q^5 - z^9 * q^8 - z^7 * q^9	\( q - \zeta_{60}^{23} q^{2} - \zeta_{60}^{16} q^{4} + \zeta_{60}^{4} q^{5} - \zeta_{60}^{9} q^{8} - \zeta_{60}^{7} q^{9} - \zeta_{60}^{27} q^{10} - \zeta_{60}^{8} q^{13} - \zeta_{60}^{2} q^{16} + ( - \zeta_{60}^{17} + \zeta_{60}^{6}) q^{17} - q^{18} - \zeta_{60}^{20} q^{20} + \zeta_{60}^{8} q^{25} - \zeta_{60} q^{26} + (\zeta_{60}^{19} + \zeta_{60}^{3}) q^{29} + \zeta_{60}^{25} q^{32} + ( - \zeta_{60}^{29} - \zeta_{60}^{10}) q^{34} + \zeta_{60}^{23} q^{36} + ( - \zeta_{60}^{25} + \zeta_{60}^{9}) q^{37} - \zeta_{60}^{13} q^{40} + ( - \zeta_{60}^{28} + \zeta_{60}^{21}) q^{41} - \zeta_{60}^{11} q^{45} + \zeta_{60}^{20} q^{49} + \zeta_{60} q^{50} + \zeta_{60}^{24} q^{52} + (\zeta_{60}^{22} - \zeta_{60}^{5}) q^{53} + ( - \zeta_{60}^{26} + \zeta_{60}^{12}) q^{58} + (\zeta_{60}^{29} + \zeta_{60}^{27}) q^{61} + \zeta_{60}^{18} q^{64} - \zeta_{60}^{12} q^{65} + ( - \zeta_{60}^{22} - \zeta_{60}^{3}) q^{68} + \zeta_{60}^{16} q^{72} + ( - \zeta_{60}^{19} + \zeta_{60}^{17}) q^{73} + ( - \zeta_{60}^{18} + \zeta_{60}^{2}) q^{74} - \zeta_{60}^{6} q^{80} + \zeta_{60}^{14} q^{81} + ( - \zeta_{60}^{21} + \zeta_{60}^{14}) q^{82} + ( - \zeta_{60}^{21} + \zeta_{60}^{10}) q^{85} + (\zeta_{60}^{26} + \zeta_{60}^{5}) q^{89} - \zeta_{60}^{4} q^{90} + (\zeta_{60}^{28} - \zeta_{60}^{4}) q^{97} + \zeta_{60}^{13} q^{98} +O(q^{100}) \) q - z^23 * q^2 - z^16 * q^4 + z^4 * q^5 - z^9 * q^8 - z^7 * q^9 - z^27 * q^10 - z^8 * q^13 - z^2 * q^16 + (-z^17 + z^6) * q^17 - q^18 - z^20 * q^20 + z^8 * q^25 - z * q^26 + (z^19 + z^3) * q^29 + z^25 * q^32 + (-z^29 - z^10) * q^34 + z^23 * q^36 + (-z^25 + z^9) * q^37 - z^13 * q^40 + (-z^28 + z^21) * q^41 - z^11 * q^45 + z^20 * q^49 + z * q^50 + z^24 * q^52 + (z^22 - z^5) * q^53 + (-z^26 + z^12) * q^58 + (z^29 + z^27) * q^61 + z^18 * q^64 - z^12 * q^65 + (-z^22 - z^3) * q^68 + z^16 * q^72 + (-z^19 + z^17) * q^73 + (-z^18 + z^2) * q^74 - z^6 * q^80 + z^14 * q^81 + (-z^21 + z^14) * q^82 + (-z^21 + z^10) * q^85 + (z^26 + z^5) * q^89 - z^4 * q^90 + (z^28 - z^4) * q^97 + z^13 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{4} + 2 q^{5}+O(q^{10}) \) 16 * q - 2 * q^4 + 2 * q^5	\( 16 q - 2 q^{4} + 2 q^{5} - 2 q^{13} + 2 q^{16} + 4 q^{17} - 16 q^{18} + 8 q^{20} + 2 q^{25} - 8 q^{34} - 2 q^{41} - 8 q^{49} - 4 q^{52} - 2 q^{53} - 2 q^{58} + 4 q^{64} + 4 q^{65} + 2 q^{68} + 2 q^{72} - 6 q^{74} - 4 q^{80} - 2 q^{81} - 2 q^{82} + 8 q^{85} - 2 q^{89} - 2 q^{90}+O(q^{100}) \) 16 * q - 2 * q^4 + 2 * q^5 - 2 * q^13 + 2 * q^16 + 4 * q^17 - 16 * q^18 + 8 * q^20 + 2 * q^25 - 8 * q^34 - 2 * q^41 - 8 * q^49 - 4 * q^52 - 2 * q^53 - 2 * q^58 + 4 * q^64 + 4 * q^65 + 2 * q^68 + 2 * q^72 - 6 * q^74 - 4 * q^80 - 2 * q^81 - 2 * q^82 + 8 * q^85 - 2 * q^89 - 2 * q^90

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1300.1.cy.a

Level	$1300$
Weight	$1$
Character orbit	1300.cy
Analytic conductor	$0.649$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{60}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.648784516423\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\Q(\zeta_{60})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{60}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{60} + \cdots)\)

\(n\)	\(301\)	\(651\)	\(677\)
\(\chi(n)\)	\(-\zeta_{60}^{25}\)	\(-1\)	\(-\zeta_{60}^{21}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 123.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
325.bi	even	60	1	inner
1300.cy	odd	60	1	inner

$p$	$F_p(T)$
$2$	\( T^{16} + T^{14} + \cdots + 1 \) T^16 + T^14 - T^10 - T^8 - T^6 + T^2 + 1
$3$	\( T^{16} \) T^16
$5$	\( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{2} \) (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^2
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( (T^{8} + T^{7} - T^{5} + \cdots + 1)^{2} \) (T^8 + T^7 - T^5 - T^4 - T^3 + T + 1)^2
$17$	\( T^{16} - 4 T^{15} + \cdots + 1 \) T^16 - 4T^15 + 11T^14 - 16T^13 + 16T^12 + 2T^11 + 8T^10 - 58T^9 + 177T^8 - 206T^7 + 52T^6 + 176T^5 - 204T^4 + 28T^3 + 89T^2 - 12*T + 1
$19$	\( T^{16} \) T^16
$23$	\( T^{16} \) T^16
$29$	\( T^{16} + T^{14} + \cdots + 1 \) T^16 + T^14 - 20T^12 - 46T^10 + 114T^8 + 409T^6 + 520T^4 - 4T^2 + 1
$31$	\( T^{16} \) T^16
$37$	\( T^{16} - 8 T^{14} + \cdots + 1 \) T^16 - 8T^14 + 30T^12 - 38T^10 - 21T^8 + 58T^6 + 35T^4 + 3*T^2 + 1
$41$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 - T^14 + 2T^13 + 16T^12 + 14T^11 + 32T^10 + 56T^9 + 42T^8 + 148T^7 + 283T^6 + 188T^5 + 36T^4 + 76T^3 + 86T^2 - 6T + 1
$43$	\( T^{16} \) T^16
$47$	\( T^{16} \) T^16
$53$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 - 3T^14 - 12T^13 - 8T^12 + 10T^11 + 33T^10 + 24T^9 + 15T^8 + 36T^7 - 87T^6 - 20T^5 + 142T^4 - 48T^3 + 87T^2 + 8*T + 1
$59$	\( T^{16} \) T^16
$61$	\( T^{16} - 3 T^{14} + \cdots + 1 \) T^16 - 3T^14 - 10T^12 + 32T^10 + 69T^8 + 8T^6 + 30T^4 - 7*T^2 + 1
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( T^{16} - 2 T^{14} + \cdots + 1 \) T^16 - 2T^14 + 13T^12 - 49T^10 + 150T^8 + T^6 + 523T^4 - 37T^2 + 1
$79$	\( T^{16} \) T^16
$83$	\( T^{16} \) T^16
$89$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 - 3T^14 - 6T^13 + 13T^12 + 16T^11 - 36T^10 + 12T^9 + 153T^8 + 174T^7 + 186T^6 + 118T^5 + 28T^4 - 12T^3 + 18T^2 - 4*T + 1
$97$	\( (T^{8} + 10 T^{5} + \cdots + 25)^{2} \) (T^8 + 10T^5 - 5T^4 + 25T^2 - 25T + 25)^2
show more
show less